Efficient Estimation in Single Index Models through Smoothing splines

TL;DR

This paper introduces a smoothing spline-based method for estimating single index models, providing theoretical guarantees and demonstrating effectiveness through simulations and real data analysis.

Contribution

It develops a novel smoothing spline approach for single index models, establishing consistency, asymptotic normality, and efficiency of estimators.

Findings

Estimators are consistent and asymptotically normal.

Method achieves asymptotic efficiency under homoscedastic errors.

Finite sample simulations support theoretical results.

Abstract

We consider estimation and inference in a single index regression model with an unknown but smooth link function. In contrast to the standard approach of using kernels or regression splines, we use smoothing splines to estimate the smooth link function. We develop a method to compute the penalized least squares estimators (PLSEs) of the parametric and the nonparametric components given independent and identically distributed (i.i.d.)~data. We prove the consistency and find the rates of convergence of the estimators. We establish asymptotic normality under under mild assumption and prove asymptotic efficiency of the parametric component under homoscedastic errors. A finite sample simulation corroborates our asymptotic theory. We also analyze a car mileage data set and a Ozone concentration data set. The identifiability and existence of the PLSEs are also investigated.